Lotka-Volterra (Exploitation / Predator-Prey):

\[\frac{\mathrm{d}}{\mathrm{d}t}x_i = x_i \left(r_i + \sum_j a_{i,j} x_j \right) \]

• \(x_i\): density of species \(i\)
• parentheses: per capita rate of increase
• \(r_i\): growth rate
• \(a_{i,j}\): effect of species \(j\) on species \(i\)

• Size structure induces \(\perp\) food webs.

• Separation of time-scales.

• Strictly linear response.

Add dispersal to see how communities interact.

\[\frac{\mathrm{d}}{\mathrm{d}t}x_i = x_i \left(r_i + \sum_j a_{i,j} x_j \right) + \sum_j d_{i,j} x_j \] - \(d_{i,j}\): \(>0\) if \(i\) and \(j\) are the same species on different islands and can move between islands. \(<0\) if \(i = j\) to conserve mass.

##    DatasetID Productivity1 Productivity2    ProdDiff      Winner
## 1       14 1     125.42370      126.7982  -1.3745517      Hybrid
## 2       19 1     162.73580      160.1915   2.5443294 Community 1
## 3       20 1     122.38829      130.3306  -7.9423479      Hybrid
## 4        5 2     266.12522      266.2753  -0.1501297      Hybrid
## 5        5 2     266.12522      282.6293 -16.5040712      Hybrid
## 6        5 2     266.27535      282.6293 -16.3539416      Hybrid
## 7       14 2      93.96649      123.9299 -29.9633981 Community 2
## 8       17 2     167.14026      182.0757 -14.9354200      Hybrid
## 9       20 2     116.32796      109.0136   7.3143753 Community 1
## 10      26 2     177.09200      182.1006  -5.0086387      Hybrid
## 11      27 2      32.11787       36.9325  -4.8146284      Hybrid
## 12      30 2     151.97244      138.8907  13.0817663      Hybrid
## 13      33 2     117.20331      183.8896 -66.6863357 Community 1
## 14       2 3     172.97277      150.5365  22.4362595      Hybrid
## 15       6 3     196.10556      230.7470 -34.6413960 Community 2

Uninvadability to assembly is not the same as uninvadability to dispersal.

Primary productivity does not seem like a great predictor.

When a hybrid community emerges, it seems to propagate to all islands, even if it does not persist.

Hybrid communities that do persist seem to be invadable.

Currently:

• Working on testing explicitly if the mass effects \(\implies\) invading the uninvadable.
• Looking for indirect mutualisms/competitions (so far, former not apparent).
• Modifying the pool (less connected, more mutual/competitive, self-regulation).

Later:

• Include dynamics on the pool.
• Integrate with evolutionary processes.
• Add more spatial dimension to the islands.